How Do I Sketch Complex Regions in the Complex Plane?

[Polamaluisraw]

Homework Statement

I do not have specific problem, I am struggling in my complex variables class and I think a large part of it is because I struggle at sketching regions in ℂ.


For instance let z=x+

I full understand what |z|< 1 looks like and all that (punctured disk, things in that form)
but when I am asked to sketch a set such as {z = x+iy : x = y, |z| < 1} then I completely fall apart.

here is a problem from a textbook,
Let Ω be the unit disk |z| < 1. Sketch sets Ω and f(Ω) where w=f(z)=z-i. Likewise where
f(z)=2z+3i.

I understand the unit disk but then after that I fall apart completely.

Thank you very much

 

[Some Pig]

Ω:|z|<1 is the open unit circular disc, center at the origin.
f(Ω) for f(z)=z-i, translate Ω to -i. The unit circular disc, center at -i.
f(Ω) for f(z)=2z+3i, dilation by 2, then translate to 3i. The circular disc with radius 2, center at 3i.